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Quadratic Expressions and Equations
Concept of Quadratic Expressions 1. A quadratic expression 'is an expression of the form '''ax² + bx + c where a, b and c are constants or real numbers, a ' = 0 and x is an unknown.' 2. A quadratic expression is a second-degree expression because the highest power of the variable x is 2 (which is x²) 3. When two linear expressions are multiplied together, the product is a '''quadratic expression'. Such a process is also known as expansion '''of two linear expressions. 4. The expansion can be carried out by multiplying each term in the first expression with every term in the second expression. 5. The expansion can be crried out by '''using a table. 6. Instead of using tables, the expansion can also be carried out by directly multiplying the terms. Factorisation of Quadratic Expressions 1. ax² + bx + c is a quadratic expression with three terms. Factorising '''this quadratic expression means expressing the sum of the three terms as a product of two linear expressions. It is the reverse operation of expansion. 2. The factorised form of a quadratic expression is the simplest form of the expression. If the quadratic expression cannot be factorised, then the expression is already in its simplest form. 3. In factorising quadratic expressions of the form ax (power 2) + bx + c where b = 0 or c = 0, we have to factorise by finding the common factors. Thus, we factorise an expression by following the steps below: Step 1: Find the highest common factor (HCF). Step 2: Place the HCF in front of the bracket. Step 3: Place the remaining factors inside the brackets after dividing each term with the HCF. 4. '''When b = 0, there is no "x" term and the resulting expression is of the form ax (power 2) + c. 5. When c = 0, there is no constant term and the resulting expression is of the form ax (power 2) + bx. 6. In factorising quadratic expressions of the form px² - q where p and q are perfect squares, we can use the "difference of two squares" '''rule. 7. The "difference of two squares" is the rule where we multiply the sum of the numbers x and y by their difference. 8. '''Factorising quadratic expressions of the form px² - q where p and q are perfect squares '''using the "difference of two squares" rule. 9. For expressions ax² + bx + c where all coefficients are non-zero (a = 0, b = 0 and c = 0), the method used to factorise an expression x² + bx + c depends on the values of b and c. 10. The following situations will assist students to determine with confidence '''when to use negative or positive integers to factorise x² + bx + c. 11. Since the trail and error method used to factorise x² + bx + c is tedious and long-winded, we can factorise by inspection. 12. We can also use the cross method to factorise quadratic expressions in the form of x² + bx + c. 13. The two factors obtained using the cross method are interchangeable. 14. The cross method is a convenient way to factorise quadratic expressions. 15. The factorisation process of ax² + bx + c in which a ''' = 1 is similar to that x² + bx + c. However, we have to consider not only the factors of c but also the factors of a.' 16. When factorising quadratic expressions containing coefficients with common factors, we extract the common factors first before factorising. 17. There are quadratic expressions which cannot be factorised. Concept of Quadratic Equations 1. A '''quadratic equation in one unknown '''is an equation of the form '''ax² + bx + c = 0 where a, b and c are real numbers with a ' = 0'.' 2. A quadratic equation '''is an equation of the second degree in variable x (where the highest power of x is two). 3. The '''general form '''of a quadratic equation is '''ax² + bx + c = 0. Roots of Quadratic Equations 1. The root of a quadratic equation is a number that when substituted for the variable in the quadratic equation, satisties the equations that is, it makes both sides equal. It is known as the solution of the equation. 2. In order to determine wheater a given value is the root of a quadratic equation, we have to substituted the given value into the equation. If the value satisfies the equation, then the value is the root of the equation. 3. Most quadratic equations have two unequal real roots but some quadratic equations have two unequal real roots but some quadratic equations have two equal real roots which is only one real root while others have no real roots (the equation x² = -9 has no roots). 4. Hence, we say that a quadratic equation can have two roots '''or '''only one root '''or '''none at all. 5. Solving quadratic equations by trail and error method: When we solve a quadratic equation, we are actually finding the value of the unknown (root) in the equation. When solving equation with the trail and error method, we substitute values for the unknown into the equation to get two values that satisfy the equation. 6. When using the trail and error method to solve x² + bx + c = 0, we shall substitute the unknown with the factors of c (positive or negative) taking into consideration of b so as to eliminate too many combinations. 7. Solving quadratic equations by factorisation: One of the most valueable applications of factorisation is in the solution of quadratic equations. We must factorise the left-hand side (LHS) of the equation, then solve the equation. For this method to work, the right-hand side (RHS) must be equal to zero. 8. ± means there are two solutions, one positive and one negative. 9. There is no standard method of solution. Students must study the facts given, tranlate them into mathematical form, solve the resulting equation and then check their answer. Vocab Corner 1. quadratic expression - ungkapan kuadratik 2. real number - nombor nyata 3. highest power - kuasa tertinggi 4. linear expression - ungkapan linear 5. expansion - pengembangan 6. factorising - pemfaktoran 7. simplest form - bentuk termudah 8. common factor - faktor sepunya 9. highest common factor - faktor sepunya terbesar 10. perfect square - kuasa dua sempunya 11. difference of two squares - beza antara dua kuasa dua 12. coefficient - pekali 13. trail and error - cuba-cuba 14. inspection - pemerinyuan 15. cross method - kaedah darab silang 16. quadratic equation in one unknown - persamaan kuadratik dalam satu anu 17. general form - bentuk am 18. root - punca 19. substituted - digantikan 20. satisfies - memuaskan 21. solution - penyelesaian 22. standard method - kaedah piawai